In Section  we will treat the case where \(O\cong {\mathbb {S}}^{2n}\)  and \(Y\cong \mathbb {S}^{2n}\)  or \(Y\cong \mathbb {S}^n\times \mathbb {S}^n\) , with \(d\ge 2\)  and \(r\ge 1\) . A result of Kreck [1]} yields that \(\mathcal {C}_{d,r}(O,Y)\)  has the cancellation property, while the inclusion and intersection properties are proved in Appendix B, building up on results of Kreck [2]}. Next, a theorem of Sullivan [3]} implies that the mapping class group of every suited submanifold of \(\mathcal {C}_{d,r}(O,Y)\)  is of type \(F_\infty \) ; see Proposition REF  below. Moreover, the complex of pieces has the correct connectivity properties, by a version of a theorem of Galatius–Randal-Williams [4]} proved in Appendix . In particular, we will obtain:
